Icarus 231 (2014) 273–286

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Icarus

journal homepage: www.elsevier.com/locate/icarus

Atlas of three body mean motion resonances in the Solar System

Tabaré Gallardo

Departamento de Astronomía, Instituto de Física, Facultad de Ciencias, Universidad de la República, Iguá 4225, 11400 Montevideo, Uruguay article info abstract

Article history: We present a numerical method to estimate the strengths of arbitrary three body mean motion reso- Received 22 July 2013 nances between two planets in circular coplanar orbits and a massless particle in an arbitrary orbit. This Revised 14 December 2013 method allows us to obtain an atlas of the three body resonances in the Solar System showing where are Accepted 17 December 2013 located and how strong are thousands of resonances involving all the planets from 0 to 1000 au. This atlas Available online 27 December 2013 confirms the dynamical relevance of the three body resonances involving Jupiter and Saturn in the aster- oid belt but also shows the existence of a family of relatively strong three body resonances involving Ura- Keywords: nus and Neptune in the far Trans-Neptunian region and relatively strong resonances involving terrestrial Asteroids, dynamics and jovian planets in the inner planetary system. We calculate the density of relevant resonances along Celestial mechanics Resonances, orbital the Solar System resulting that the main asteroid belt is located in a region of the planetary system with Kuiper belt the lowest density of three body resonances. The method also allows the location of the equilibrium points showing the existence of asymmetric librations (r – 0 or 180). We obtain the functional depen- dence of the resonance’s strength with the order of the resonance and the eccentricity and inclination of the particle’s orbit. We identify some objects evolving in or very close to three body resonances with Earth–Jupiter, Saturn–Neptune and Uranus–Neptune apart from Jupiter–Saturn, in particular the NEA 2009 SJ18 is evolving in the resonance 1 1E 1J and the centaur 10199 Chariklo is evolving under the influence of the resonance 5 2S 2N. Ó 2013 Elsevier Inc. All rights reserved.

1. Introduction critical angles, identified thousands of asteroids in TBRs with Jupi- ter and Saturn, concluding there are more asteroids in TBRs than in Fifteen years ago, concentrated over a period of about a year, a two body resonances. succession of papers were published (Murray et al., 1998; Nesv- The approximate nominal position in semimajor axis of the orny´ and Morbidelli, 1998, 1999; Morbidelli and Nesvorny´ , 1999) TBRs taking arbitrary pairs of planets is very simple if we ignore showing an intense theoretical and numerical work on three body the secular motion of the perihelia and nodes of the three bodies. mean motion resonances (TBRs) involving an asteroid and two When these slow secular motions are taken into account each of massive planets. These papers, that have their roots on earlier the nominal TBRs split in a multiplet of resonances all them very works devoted to the study of an asteroid in zero order TBRs near the nominal one (Morbidelli, 2002). The challenge is to obtain (Wilkens, 1933; Okyay, 1935; Aksnes, 1988), stated the relevance the strength, width or libration timescale that give us the dynam- that the TBRs involving Jupiter–Saturn and also Mars–Jupiter have ical relevance of these resonances. Analytical planar theories in the long term stability in the asteroid’s region. In spite of being developed by Murray et al. (1998) and Nesvorny´ and Morbidelli weaker than the two body resonances they are much more numer- (1999) allowed to describe and understand the dynamics of the ous generating several dynamical features in the asteroidal popu- TBRs involving Jupiter and Saturn in the asteroidal region. These lation, like concentrations for some values of semimajor axes, theories are appropriated to study in detail specific resonances anomalous amplitude librations and chaotic evolutions (Nesvorny´ with Jupiter and Saturn but its application to any arbitrary reso- and Morbidelli, 1998). In particular, both borders of the main aster- nance involving any planet is not trivial, which possibly explains oid belt exhibit chaotic diffusion due to the superposition of the absence of papers published on this topic since these years several weak two-body and TBRs (Murray and Holman, 1997; with the exceptions of a few ones devoted to specific scenarios Morbidelli and Nesvorny´ , 1999). More recently, Smirnov and (Guzzo, 2005; de La Fuente Marcos and de La Fuente Marcos, Shevchenko (2013) in a massive numerical integration of 249,567 2008; Cachucho et al., 2010; Quillen, 2011). asteroids by 105 years and looking at the time evolution of the In this paper, from a different approach, we will obtain a global view of all dynamically relevant TBRs involving all the planets ta- ken by pairs along all the Solar System. Our method is not E-mail address: gallardo@fisica.edu.uy analytical but numerical and it is based on an estimation of the

0019-1035/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.icarus.2013.12.020 274 T. Gallardo / Icarus 231 (2014) 273–286

1-2J+1S, e=0.0 1-2J+1S, e=0.1 ) ) σ σ ( ( ρ ρ

0 90 180 270 360 0 90 180 270 360 σ σ

1-2J+1S, e=0.2 1-2J+1S, e=0.3 ) ) σ σ ( ( ρ ρ

0 90 180 270 360 0 90 180 270 360 σ σ

1-2J+1S, e=0.4 1-2J+1S, e=0.5 ) ) σ σ ( ( ρ ρ

0 90 180 270 360 0 90 180 270 360 σ σ

Fig. 1. Evolution of the shape of qðrÞ for growing eccentricities for the zero order resonance 1-2J + 1S at a ¼ 3:8045 au. Numerical integrations of test particles show that the asymmetric librations around r 90 and r 270 are stable and also exist large amplitude librations around r ¼ 180 that wrap both asymmetric libration centers. The librations around r ¼ 0 are unstable.

strength of the resonances which is obtained evaluating the effects is oscillating over time, being ki the mean longitudes and ki integers. of the mutual perturbations in all possible spatial configurations of More precisely, for massless particles with inclined orbits there are the three bodies satisfying the resonant condition. It allows us to other possible definitions for the critical angle involving combina- appreciate the effect of arbitrary eccentricity and inclination of tions of its X0 and -0, but it is possible to show that after an appro- the massless particle’s orbit on the resonance’s strength but, in or- priate averaging procedure the leading term in the expansion of the der to reduce the number of parameters involved in the problem, resulting disturbing function will be the one whose argument is gi- we impose coplanar and circular orbits for the two perturbing ven by Eq. (1) (Nesvorny´ and Morbidelli, 1999). In this work we will planets. Nevertheless, the method can be extended to arbitrary use the following notation: the order of the resonance is planetary orbits. In the next section we start describing our numer- q ¼jk0 þ k1 þ k2j, we call p ¼jk0jþjk1jþjk2j and we note ical method and we explore how the calculated strengths depend k0 þ k1P1 þ k2P2 the resonance involving planets P1 and P2. The on the parameters of the problem. In Section 3 we analyze the approximate nominal location of the resonance assuming unper- distribution of TBRs along the Solar System and discuss its turbed Keplerian motions is dynamical effects providing some examples. In Section 4 we qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi present the conclusions. 3=2 k1 3 k2 3 a0 ’ ð1 þ m1Þ=a1 ð1 þ m2Þ=a2 ð2Þ k0 k0 2. Numerical approximation to the disturbing function for three body resonances where ai and mi are the mean semimajor axes of the planets and its masses expressed in solar masses respectively. In the real Solar Sys- Three body resonances between a massless particle with an tem the actual location depends on the precession of the perihelia arbitrary orbit given by ða0; e; i; X0; -0Þ and two planets P1 and P2 and the gravitational effects of the other planets not taken into in circular coplanar orbits occur when the critical angle account. In the very far Trans-Neptunian Region (TNR), depending on the resonance, the actual location could be shifted something r ¼ k k þ k k þ k k ðk þ k þ k Þ- ð1Þ 0 0 1 1 2 2 0 1 2 0 between 0.1 au and 1 au. T. Gallardo / Icarus 231 (2014) 273–286 275

1-3J+1S, e=0.01 1-3J+1S, e=0.1 ) ) σ σ ( ( ρ ρ

0 90 180 270 360 0 90 180 270 360 σ σ

1-3J+1S, e=0.2 1-3J+1S, e=0.3 ) ) σ σ ( ( ρ ρ

0 90 180 270 360 0 90 180 270 360 σ σ

1-3J+1S, e=0.4 1-3J+1S, e=0.5 ) ) σ σ ( ( ρ ρ

0 90 180 270 360 0 90 180 270 360 σ σ

Fig. 2. Evolution of the shape of qðrÞ for growing eccentricities for the first order resonance 1-3J + 1S at a ¼ 2:7518 au, the resonance where is evolving 485 Genua. For large eccentricities asymmetric equilibrium points appear, which are stable according to numerical integrations of test particles. There are also large amplitude librations around r ¼ 180 that wrap both asymmetric libration centers.

2.758

2.756

2.754

2.752

2.75 mean a (au)

2.748

360

270

180 (1-3J+1S)

σ 90

0 0 5000 10000 15000 20000 25000 30000 35000 40000 time (yrs)

Fig. 3. A fictitious particle with 0:3 < e < 0:4 and i < 4 evolving in the real planetary system with temporary captures in the asymmetric equilibrium points of the resonance 1-3J + 1S. In top panel the mean semimajor axis calculated as the mean value in a running window of 1000 yrs. The horizontal line indicates the nominal resonance. Bottom panel: the time evolution of the critical angle r ¼ k 3kJ þ kS þ -. 276 T. Gallardo / Icarus 231 (2014) 273–286

360

270

σ 180

90

0 0 5000 10000 15000 20000 25000 30000 time (yrs)

Fig. 4. The critical angle for a fictitious particle with e 0:5 and i ¼ 0 captured in an asymmetric equilibrium point in the resonance 1-3J + 1S assuming Jupiter and Saturn in circular coplanar orbits.

0.1 Table 1 The strongest resonances with Dq > 0:001 in intervals of 0.05 au from 0 to 4 au 0.01 calculated assuming e ¼ 0:15; i ¼ 6; x ¼ 60.

a (au) Resonance q Dq 0.001 0.7946 8 7V þ 1J 2 0.00448 1.0980 7 6E 1J 0 0.00450 0.0001 Δρ 1.1020 8 7E þ 1J 2 0.00238 1.9389 1 6J þ 4S 1 0.00131 1e-005 2.1376 1 5J þ 3S 1 0.00260 2.3031 1 5J þ 4S 0 0.00121 2.3967 1 4J þ 2S 1 0.00288 1e-006 2.4493 2 9J þ 7S 0 0.00127 2.5026 2 8J þ 5S 1 0.00151 1e-007 2.6211 1 4J þ 3S 0 0.00282 1e-011 1e-010 1e-009 1e-008 1e-007 2.6845 2 7J þ 4S 1 0.00317 β 2.7518 1 3J þ 1S 1 0.00161 2.8266 2 7J þ 5S 0 0.00573 Fig. 5. The semiamplitude Dq calculated with Eq. (22) assuming e ¼ 0:1; i ¼ 0 as a 2.8506 3 9J þ 4S 2 0.00107 function of the semiamplitude b deduced from Nesvorny´ and Morbidelli (1999) for 2.9034 2 6J þ 3S 1 0.00538 the 19 resonances of their Table 1. The curve fitting corresponds to Dq 3 105b. 2.9587 3 9J þ 5S 1 0.00248 3.0155 3 8J þ 3S 2 0.00171 According to Nesvorny´ and Morbidelli (1999), in the planar 3.0777 1 3J þ 2S 0 0.02678 problem the Hamiltonian for the particle can be expressed in a 3.1406 3 8J þ 4S 1 0.00489 3.1732 2 5J 2S 1 0.00288 simplified form as depending on canonical variables that depend þ ffiffiffi 3.2794 3 8J þ 5S 0 0.00614 3=2p on ða; rÞ. The width in au of the resonance is Da / a b where 3.3336 4 9J þ 3S 2 0.00241 b is the semi-amplitude of the resonant disturbing function, that 3.3933 2 5J þ 3S 0 0.01548 we note as RðrÞ. After a thorough analytical procedure they ob- 3.4160 5 11J þ 4S 2 0.00251 tained an analytical expression for RðrÞ for the planar eccentric 3.4532 4 9J þ 4S 1 0.00600 3.5176 3 7J þ 4S 0 0.01068 case asteroid–Jupiter–Saturn allowing them to obtain analytical 3.5842 4 9J þ 5S 0 0.00556 solutions. In this paper we are looking for a numerical approxima- 3.6059 3 6J þ 2S 1 0.00562 tion to RðrÞ for massless bodies in arbitrary orbits in resonance 3.6822 5 10J þ 4S 1 0.00619 with arbitrary pairs of planets considered in coplanar circular or- 3.7408 5 9J þ 2S 2 0.00490 3.7513 6 11J 3S 2 0.00219 bits. This approximation will help us to understand which they þ 3.8045 1 2J þ 1S 0 0.02768 are and how are distributed the dynamically relevant TBRs in the 3.8846 4 7J þ 2S 1 0.01427 Solar System. The algorithm devised here can be extended to 3.9123 3 5J þ 1S 1 0.01196 eccentric planetary orbits and applied to other planetary systems. 3.9727 4 8J þ 5S 1 0.00549 The mean resonant disturbing function, RðrÞ, that drives the resonant motion of the particle could be ideally calculated elimi- where k0 was explicitly written in terms of the variables k1; k2 and nating the short period terms of the resonant disturbing function the parameters r; -0 using Eq. (1) and where Rðk0; k1; k2Þ¼

R by means of R01 þ R02 being Z 1 T ~ ~ ~ ~ ~ 2 1 ri rj RðrÞ¼ Rðr0ðtÞ; r1ðtÞ; r2ðtÞÞdt ð3Þ Rij ¼ k mjð Þð5Þ T 0 3 rij rj where T is an ideal interval, that means long enough for the system where k is the Gaussian constant and ~r ; ~r are the heliocentric posi- to be evaluated in all possible configurations of the heliocentric i j tions of bodies with subindex i and j respectively. Note that for each positions, ~ri, of the three bodies but not long enough to appreciate set of values ; k ; k ; there are k values of k that satisfy Eq. changes in r. If we can admit the use of the Keplerian unperturbed ðr 1 2 -0Þ 0 0 (1), which are: positions then we can substitute the integral in time domain by the integral in phase space: k ¼ ðÞr k k k k þðk þ k þ k Þ- =k þ n2p=k ð6Þ Z Z 0 1 1 2 2 0 1 2 0 0 0 1 2p 2p k k ; k ; k ; ; k ; k k with n ¼ 0; 1; ...; k0 1. All them contribute to RðrÞ in Eq. (4) so we RðrÞ¼ 2 d 1 RðÞ0ðr 1 2 -0Þ 1 2 d 2 ð4Þ 4p 0 0 have to evaluate all these k0 terms and calculate the mean, which is T. Gallardo / Icarus 231 (2014) 273–286 277

0 1

0.01 q=0 -1 0.0001 -2 q=1 1e-006

) q=2 Δρ -3 1e-008 Δρ q=3 log ( -4 1e-010

1e-012 -5 q=4 1e-014 0.01 0.1 -6 -7 -6 -5 -4 -3 -2 -1 0 eccentricity log (SR) Fig. 9. Continuous lines: Dq as function of the asteroid’s eccentricity for zero (q ¼ 0, Fig. 6. Dq in logarithmic scale calculated with Eq. (25) for two body resonances 1–3J + 2S at 3.0777 au), first (q ¼ 1, 1–4J + 4S at 2.9067 au), second (q ¼ 2, 2–7J + 7S versus the strength SR in logarithmic scale according to Gallardo (2006) for 9 at 3.1773 au), third (q ¼ 3, 1–5J + 7S at 3.0854 au) and fourth (q ¼ 4, 1–6J + 9S at arbitrary two body resonances assuming a circular orbit for the planet and 2.9134 au) order resonances assuming i ¼ 0 . At low eccentricities these curves q e ¼ 0:1; i ¼ 0 for the particle. The curve fitting corresponds to Dq / SR0:8. correspond approximately with Dq / e whose representations are showed with dashed lines. The depression of the curve corresponding to the resonance 1–4J + 4S at e 0:05 is related to the appearance and disappearance of asymmetric equilibrium points. 0.01

0.0001 1 1e-006 0.01 q=0 1e-008 0.0001 q=2 1e-010 Δρ 1e-006 q=1

1e-012 1e-008 Δρ q=4 1e-014 1e-010

1e-016 1e-012 q=3 1e-018 1e-014 0 1 2 3 4 5 6 7 8 9 10 11 12 13 1e-016 order q 0.1 1 sin (i) Fig. 7. Dq versus order q for TBRs of order q 13 and p 20 with Jupiter and Saturn from 2 to 2.4 au, calculated assuming e ¼ 0:1 and i ¼ 0 . It is approximately Fig. 10. Continuous lines: Dq as function of the asteroid’s inclination for the same verified that lgðDqÞ/q. resonances of Fig. 9 but assuming e ¼ 0 and x ¼ 0 . For low inclinations these curves correspond approximately with Dq /ðsin iÞz where z ¼ q for even order and z ¼ 2q for odd order resonances. Dashed lines: curve fitting to ðsin iÞq for even order 1 resonances and to ðsin iÞ2q for odd order resonances.

0.01 (1968) for the study of the Hildas, extended to more variables to 0.0001 integrate as in Thomas and Morbidelli (1996) and Michtchenko 1e-006 and Malhotra (2004) but with a resonant condition like in Gomes et al. (2005). 1e-008 As we know, the disturbing function of a TBR is a second order Δρ 1e-010 function of the planetary masses, which means the calculation of the double integral (4) cannot be done over the perturbing function 1e-012 evaluated at the unperturbed heliocentric positions. This can be shown considering that R ¼ R þ R and as there is no commen- 1e-014 01 02 surability between the particle and the planet P1 nor between

1e-016 q=0 the particle and the planet P2, the mean of R01 and R02 become q=4 q=8 independent of r, then they do not contribute to RðrÞ. To properly 1e-018 2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4 evaluate the integral it is necessary to take into account their mu- ~ a (au) tual perturbations in the position vectors ri. Two body mean mo- tion resonances are a simpler case because being a first order Fig. 8. Dq versus semimajor axis for TBRs of order q 13 and p 20 with Jupiter perturbation in the planetary masses the position vectors can be and Saturn from 2 to 4 au, calculated assuming e ¼ 0:1 and i ¼ 0 where resonances substituted by the Keplerian, non-perturbed positions. of order 0, 4 and 8 are indicated with different symbols for comparison. Zero order In previous works (Murray et al., 1998; Nesvorny´ and Morbidel- resonances and resonances located closer to the planets tend to be stronger. li, 1999) analytical methods were developed in order to obtain the solution of an asteroid in TBR with Jupiter and Saturn. In this paper equivalent to integrate in k0 maintaining the condition (1).Upto we are mainly interested not in obtaining the solution but in esti- this point, this scheme is analogue to the one proposed by Schubart mate the comparative strength of thousands of resonances with all 278 T. Gallardo / Icarus 231 (2014) 273–286 the planets along all the Solar System. Faced to this problem, in or- Integrating twice we obtain the displacements with respect to the der to estimate the behavior of RðrÞ we adopt the following Keplerian motion: scheme for computing the double integral of Eq. (4): 2 ~ ðDtÞ Dr0 ’ðr0R01 þ r0R02Þ ð17Þ Rðk0; k1; k2Þ’Ru þ DR ð7Þ 2 where Ru stands from R calculated at the unperturbed positions of 2 the three bodies and DR stands from the variation in R generated ~ ðDtÞ u Dr1 ’ r1R12 ð18Þ by the perturbed (not Keplerian) displacements of the three bodies 2 in a small interval Dt. More clearly, given any set of the three posi- 2 ~ ðDtÞ tion vectors ri satisfying Eq. (1) we compute the mutual perturba- D~r ’ r R ð19Þ 2 2 21 2 tions of the three bodies and calculate the D~ri that they generate in a small interval Dt and the DR associated. This scheme is equiv- As the integral of Ru ¼ R01 þ R02 becomes independent of r, we are alent to evaluate the integral over the infinitesimal trajectory the only interested in computing the function qðrÞ defined by system follows due to the mutual perturbations when released at Z Z all possible unperturbed positions that verify Eq. (1). We have then 1 2p 2p k k qðrÞ¼ 2 d 1 DRd 2 ð20Þ 4p 0 0 Ru ¼ R01 þ R02 ð8Þ

DR ¼ DR01 þ DR02 ð9Þ always satisfying Eq. (1), being  where R01 and R02 refer to the disturbing functions evaluated at the 2 ðDtÞ 2 2 DR ¼ ðr0R01Þ þðr0R02Þ þ 2r0R01r0R02 þ r1R01r1R12 þ r2R02r2R21 unperturbed positions and DR01 and DR02 refer to the variations due 2 to displacements caused by the mutual perturbations: ð21Þ ~ ~ DR01 ¼ r0R01Dr0 þ r1R01Dr1 ð10Þ It is possible to show that the first two terms are independent of r then DR02 ¼ r0R02Dr~0 þ r2R02Dr~2 ð11Þ Z ðDtÞ2 1 2p where D~r refers to displacements with respect to the heliocentric qðrÞ¼ dk i 2 4p2 1 Keplerian motion and being Z 0 2p

2 ~rj ~ri ~rj ðÞ2r0R01r0R02 þ r1R01r1R12 þ r2R02r2R21 dk2 r R ¼ k m ð Þð12Þ 0 i ij j r3 r3 ij j ð22Þ

~r ~r ~r ~r Note that qðrÞ/m1m2 while in the case of two body resonances R k2m i j i 3 ~r ~r j 13 rj ij ¼ jð 3 3 þ ð i jÞ 5ÞðÞ the disturbing function is proportional to only one planetary mass rij rj rj making TBRs much weaker than two body resonances. We identify From the equations of motion we have: Dt with the permanence time in each element of the phase space € ðDk0; Dk1; Dk2Þ. If the double integral is computed dividing the D~r ¼ R þ R ð14Þ 0 r0 01 r0 02 dominium in N equal steps in k1 and N equal steps in k2 we can cal- culate the mean elapsed time Dt in the element of phase space as € D~r R 15 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ¼ r1 12 ð Þ 3 T T T Dt ¼ 0 1 2 ð23Þ N ~€ Dr2 ¼ r2R21 ð16Þ where Ti are the orbital periods. Then

1

0.1

0.01 Δρ

0.001 Strength

0.0001

1e-005 0.1 1 10 100 1000 a (au)

Fig. 11. Global view of the atlas of TBRs from 0.1 to 1000 au assuming e ¼ 0:15; i ¼ 6 ; x ¼ 60 . Color version: resonances which its most interior planet is Mercury, Venus, Earth, Mars, Jupiter, Saturn or Uranus are showed in red, green, blue, pink, black, red or green respectively. Note the strong resonances involving Uranus and Neptune at a > 250 au. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) T. Gallardo / Icarus 231 (2014) 273–286 279

1 1

0.1 0.1

0.01 0.01

0.001 0.001

0.0001 0.0001

1e-005 1e-005

1e-006 1e-006 0 0.5 1 1.5 2 2 2.5 3 3.5 4 a (au) a (au)

1 1

0.1 0.1

0.01 0.01

0.001 0.001

0.0001 0.0001

1e-005 1e-005

1e-006 1e-006 4 4.5 5 5.5 6 6 8 10 12 14 16 18 20 22 24 a (au) a (au)

1 1

0.1 0.1

0.01 0.01

0.001 0.001

0.0001 0.0001

1e-005 1e-005

1e-006 1e-006 25 30 35 40 45 50 40 50 60 70 80 90 100 a (au) a (au)

0.1 0.1

0.01 0.01

0.001 0.001

0.0001 0.0001

1e-005 1e-005

1e-006 1e-006

1e-007 1e-007 100 150 200 250 300 300 400 500 600 700 800 900 1000 a (au) a (au)

Fig. 12. Same as Fig. 11 with linear scale in semimajor axis. Qualitative comparison with Fig. 7 from Gallardo (2006) can be done.

4p2a a a ideally computed along the actual trajectories of the three bodies, Dt2 ¼ 0 1 2 ð24Þ k2N2 q is computed along infinitesimal perturbed trajectories around the Keplerian positions of the three bodies. Anyway, a strong The above algorithm is independent of the pair of independent vari- dependence of q with r is indicative of a strong resonance. On ables ki used in the integral and is independent of the order in which the double integral is evaluated. Taking N equal for all reso- the other hand, if the critical angle r does not affect q it will be nances its actual value is irrelevant; in our codes we use an arbi- indicative of a weak resonance. Also, an extreme of qðrÞ at some 4 2 r means that for that critical angle the perturbations have an trary value N ¼ 180 and we have divided qðrÞ by k mJupiter for convenience. Considering r as a constant parameter we calculate extreme, that means, there is an equilibrium point, but we cannot the integral (22) for a set of values of r between ð0; 2pÞ and we apply stability criteria deduced for R with qðrÞ because they are obtain numerically qðrÞ. As defined above, qðrÞ carries an arbitrary not the same function, in particular we cannot deduce whether multiplicative constant and is dimensionally equivalent to a dis- the equilibrium points are stable or unstable. turbing function. For several resonances we confronted the shape of qðrÞ with numerical integrations of fictitious particles and with the analyti- 2.1. Shapes of qðrÞ and asymmetric equilibrium points cal results by Nesvorny´ and Morbidelli (1999). For low eccentricity orbits the most common case is a cosine or sine function for qðrÞ We must stress that the function qðrÞ is not the resonant dis- with equilibrium points at 0 and 180 degrees as in the theory by turbing function RðrÞ, but closely related to it. While R is a mean Nesvorny´ and Morbidelli (1999), but for some cases, approximately 280 T. Gallardo / Icarus 231 (2014) 273–286

Table 2 Table 4 The strongest resonances with Dq > 0:001 in intervals of 1 au from 4 to 1000 au The strongest resonances with Dq > 0:00001 involving only terrestrial planets in calculated assuming e ¼ 0:15; i ¼ 6; x ¼ 60. intervals of 0.1 au calculated assuming e ¼ 0:15; i ¼ 6; x ¼ 60.

a (au) Resonance q Dq a (au) Resonance q Dq 4.9242 7 8J þ 1S 0 0.29596 0.4249 3 3Me þ 1V 1 0.00002 5.7240 3 3J þ 1S 1 3.37943 0.5822 1 2V þ 1E 0 0.00004 6.5919 2 1J 1S 0 0.20865 0.6885 8 8V 1E 1 0.00016 7.3022 3 1J 2S 0 0.16873 0.7949 7 3V 5E 1 0.00042 8.8460 4 1J 2S 1 0.30202 0.8341 2 1V 1E 0 0.00019 9.0482 6 þ 1J 9S 2 0.63886 0.9602 4 2V 1E 1 0.00014 10.4640 4 1J 1S 2 0.45567 1.0998 4 4E þ 1M 1 0.00017 11.1722 12 1J 7S 4 0.08574 1.1016 12 7V þ 1E 6 0.00070 12.1424 5 1J 1S 3 0.02895 1.2390 5 1V 2E 2 0.00004 13.2836 9 1J 3S 5 0.00518 1.3240 4 1V 1E 2 0.00002 14.9629 1 þ 1J 3S 1 0.00581 1.6774 11 4E 2M 5 0.00003 15.0977 3 þ 1J 4S 0 0.00601 1.9248 1 þ 1V 2E 0 0.00007 16.8947 4 1S 2U 1 0.00469 3.0554 2 þ 1V 2E 1 0.00001 17.9233 10 1J 4U 5 0.01298 18.0021 1 1J þ 6U 6 0.01544 19.0549 8 1J 1U 6 0.00881 20.6628 1 þ 1J 8U 6 0.01141 21.0578 7 1J þ 1U 7 0.04680 For example, in Figs. 1 and 2 we show the transformations of 22.1112 10 1J 1U 8 0.00529 qðrÞ for increasing eccentricities for two resonances. The equilib- 23.7521 2 þ 1J 3S 0 0.00482 26.8747 5 1J þ 8N 12 0.00128 rium point at r ¼ 0 in low eccentricity regime showed in Fig. 1 27.9823 8 1J þ 5N 12 0.00954 for the resonance 1-2J + 1S is unstable, but on the other hand the 28.5246 11 1J þ 2N 12 0.03031 librations around the asymmetric centers at r 90; 270 are sta- 29.4209 2 þ 1J 16N 13 0.00458 ble as our numerical integrations of fictitious particles confirm. In 30.9517 10 1S 4N 5 0.00198 31.8850 13 1J þ 2N 14 0.00322 the next case, referred to the resonance 1-3J + 1S where inhabits 32.8328 17 1J 1N 15 0.01268 485 Genua, for low eccentricity orbits our qðrÞ is very similar to 33.2439 15 1J þ 1N 15 0.05002 the resonant disturbing function obtained by Nesvorny´ and 34.1081 18 1J 1N 16 0.00354 Morbidelli (1999) but for e > 0:2 equilibrium points appear at 262.04 1 þ 1U 2N 0 0.01373 90 and 270 which we confirmed by numerical integrations of fic- 415.96 2 þ 1U 2N 1 0.00258 titious particles. Fig. 3 shows temporary captures of a fictitious particle evolving in the real planetary system in these asymmetric libration centers. In numerical integrations considering only two Table 3 planets in circular coplanar orbits these asymmetric librations The strongest resonances with Dq > 0:0001 involving terrestrial and jovian planets in intervals of 0.1 au from 0 to 4 au calculated assuming e ¼ 0:15; i ¼ 6; x ¼ 60. are considerably more stable as showed in Fig. 4 where only Jupiter and Saturn in coplanar circular orbits were considered as perturb- a (au) Resonance q Dq ers. The librations around r ¼ 180 for these two resonances are in 0.5838 5 7V þ 2J 0 0.00011 fact trajectories that wrap the asymmetric librations points. A care- 0.6993 1 1V 1J 1 0.00036 ful reader will find a temporary libration around r 90 for 485 0.7946 8 7V þ 1J 2 0.00448 0.8057 6 5V 2J 1 0.00054 Genua in Fig. 2 of Nesvorny´ and Morbidelli (1998) at 0.9817 3 3E 1J 1 0.00086 t 75,000 years. Analyzing several resonances we conclude that 1.0980 7 6E 1J 0 0.00450 asymmetric librations are very common and we can say that in 1.1020 8 7E þ 1J 2 0.00238 general qðrÞ is a good indicator to the existence of equilibrium 1.2415 3 2E 2J 1 0.00039 points but their stability needs to be studied case by case. Never- 1.3381 8 5E 2J 1 0.00039 1.4128 7 4E 2J 1 0.00039 theless, the asymmetric librations are easily destroyed by the per- 1.5040 2 1E 1J 0 0.00059 turbations due to the other non-resonant planets, then its 1.6703 9 8M þ 1J 2 0.00037 detection in real asteroids could be difficult. 1.7452 5 2E 2J 1 0.00032 1.8748 3 1E 2J 0 0.00056 1.9708 3 1E 1J 1 0.00019 2.2. Strengths Dq of the three body resonances 2.1681 4 1E 3J 0 0.00024 2.2712 4 1E 2J 1 0.00030 In order to obtain a relationship between q and the disturbing 2.4090 5 1E 4J 0 0.00010 2.5159 5 1E 3J 1 0.00019 function R we generated Fig. 5 comparing the semiamplitude 2.6377 3 1M 2J 0 0.00011 Dq ¼ðqmax qminÞ=2 with the semiamplitude b calculated by 2.7204 6 1E 4J 1 0.00012 Nesvorny´ and Morbidelli (1999) for the 19 resonances of their Table 1. This is a very difficult comparison because our calculations correspond to circular planetary orbits while Nesvorny´ and Mor- one third of the resonances studied, minima of qðrÞ not necessarily bidelli (1999) considered eccentric and time evolving orbits for mean stable nor maxima mean unstable. We also found very com- Jupiter and Saturn which probably is one of the sources of the evi- dent scatter in the figure. The curve fitting indicated in the figure mon that for medium to high eccentricity orbits equilibrium points 5 appear at asymmetric positions r – 0; 180. We analyzed their corresponds to Dq 3 10 b, but due to the errors associated existence and stability with numerical integrations of fictitious we can only say that b and Dq are roughly proportional. particles for several cases using EVORB (Fernández et al., 2002) We can devise another test for our Dq applying it to the case of as we show below, confirming in general their existence but with two body resonances. Following an analogue reasoning as devised compromised stability by several perturbations we did not take here for TBRs it is possible to show that in the case of a two body into account in our algorithm. resonance qðrÞ becomes T. Gallardo / Icarus 231 (2014) 273–286 281

200

180

160

140

120

100

80

60

Number of TBRs per 0.1 au 40

20

0 0 5 10 15 20 25 30 35 40 a (au)

Fig. 13. Number of TBRs with Dq > 105 per 0.1 au. Dq calculated assuming e ¼ 0:15; i ¼ 6 ; x ¼ 60 .

100

80

60

40

Number of TBRs per 0.1 au 20

0 1.5 2 2.5 3 3.5 4 4.5 a (au)

Fig. 14. Detail of Fig. 13 including the normalized distribution of the osculating semimajor axis of the asteroids from ASTORB database (dashed line). log (Strength) 2-7J+4S 2-7J+5S 2-6J+3S 3-8J+4S 3-8J+5S 1-4J+2S 1-4J+3S 1-3J+1S 2-5J+2S 1-3J+2S 2-1M 3-7J+2S

2 2.2 2.4 2.6 2.8 3 3.2 3.4 a (au)

Fig. 15. Strongest two body resonances (thin/blue lines) and TBRs (thick/red lines) with an histogram of proper semimajor axes from AstDyS. The strengths in logarithmic scale for both types of resonances were calculated assuming planets with circular orbits and the massless particle with e ¼ 0:2; i ¼ 10 ; x ¼ 60 . The sets of two body and TBRs are not in the same scale. Some of the strongest TBRs are labeled. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) 282 T. Gallardo / Icarus 231 (2014) 273–286

0.01 1-1E-1J

0.0001 9-6V-1N 7-6E-3M 8-6E-5M 7-8E+5J

1e-006 9-6V-2U 8-5V-1M Δρ

1e-008

1e-010

1e-012 0.946 0.9465 0.947 0.9475 0.948 0.9485 0.949 a (au)

Fig. 16. Strength Dq calculated taking e ¼ 0:11; i ¼ 13 ; x ¼ 173 for all TBRs with p 20 located near the semimajor axis of 2009 SJ18.

0.952 2009 SJ18

0.95

a (au) 0.948

0.946 360

270

180

(1-1E-1J) 90 σ

0

270

180

(6-2V-1N) 90 σ

0 0 1000 2000 3000 4000 5000 time (yrs)

Fig. 17. NEA 2009 SJ18 evolving in 1 1E 1J at a ¼ 0:94747 au with strength Dq ’ 3:6 104 showing librations around r ¼ 180 and transitory librations around the asymmetric libration center at r 270 (mid panel). In the low panel it is showed for comparison the critical angle related to the closest resonance to 1 1E 1J which is 9 6V 1N located at a ¼ 0:94744 au with strength Dq ’ 8 108.

Z 2 2p ðDtÞ 1 2 dependence with the proximity to the planets; resonances closer qðrÞ¼ ðr0R01Þ dk1 ð25Þ to the planets tend to be stronger as pointed out by Nesvorny´ 2 2p 0 and Morbidelli (1998). We found an analogue behavior, approxi- We calculated the semiamplitude Dq for some two body resonances mately symmetric, for the external TBRs with Uranus and Neptune and compared with the strength, SR, given by Gallardo (2006) in the TNR. Note in Fig. 8 the pattern at a 3.8 au related to the which in this case was calculated assuming a circular orbit for the superposition of two two-body resonances: 1-4S at a ¼ 3:7915 au planet. The result indicating a more clear relationship Dq / SR0:8 and 5-8J at a = 3.8021 au which added are very close to the TBR is showed in Fig. 6. With these two tests we can conclude that Dq 3-4J-2S at a = 3.8002 au. As the resonant condition (1) can also _ _ _ _ is nearly proportional to the semiamplitude of the resonant disturb- be written ðk0 þ jÞk0 þ k1k1 ¼ jk0 k2k2 ¼ m, being j an arbitrary ing function and in spite of the limitations of the method can be integer, if a0 is such that for some j it is verified m ’ 0, then the considered a rough measure of the resonance’s strength. We will TBR will be the resulting of the superposition of two two-body res- use this quantity for studying TBRs, since it is easy to compute onances and a particular arrangement of TBRs appears near a0. and a good compromise between analytical understanding and The dependence with the eccentricity and the order of the res- accuracy on one hand, and effective exploration of a vast parameter onance is showed in Fig. 9 for orbits with i ¼ 0 and the depen- space, on the other. dence with the inclination and the order of the resonance is To analyze the properties of Dq we calculated it for 489 TBRs showed in Fig. 10 for e ¼ 0 respectively. With these and others with Jupiter and Saturn with order q 6 13 and p 6 20 located be- plots going up to q ¼ 5 we deduced by curve fitting that for zero tween 2 and 4 au summarizing the results in Figs. 7 and 8. From inclination orbits Dq / eq, which is coherent with theoretical mod- Fig. 7 it is evident an exponential decay of Dq with the order q els for coplanar orbits that predict the lower order terms in the dis- as theoretical models predict and from Fig. 8 it is clear also a turbing function are factorized by eq. Moreover, we find that for T. Gallardo / Icarus 231 (2014) 273–286 283

263.55

263.54

263.53 mean a (au) 263.52

360

270

180 (1+1U-2N)

σ 90

0 0 100 200 300 400 500 0 100 200 300 400 500 600 time (Myrs) time (Myrs)

Fig. 18. Two particles with an imposed migration in semimajor axis interacting with the resonance 1 + 1U 2N in a fictitious planetary system composed only by the Sun and the planets Uranus and Neptune in coplanar quasi circular orbits. Left panels: the particle is initially outside the resonance and crossed the resonance due to the migration. Right panels: the particle is initially inside the resonance and resists the forced migration due to the resonance’s strength. The mean a is calculated with a window of 1 Myrs.

The nominal position of the resonance for this particular planetary system is a0 ¼ 264:93 au but its actual position is shifted due to the motion of the perihelia that were ignored in Eq. (2). All particles have e ¼ 0:2 and the resonance’s strength is Dq ’ 0:014.

295 290 2006 UL321 285 280 275 270 265 mean a (au) 260 255

270

180

(1+1U-2N) 90 σ

0 0 1 2 3 4 time (Myrs)

Fig. 19. Scattered Disk Object 2006 UL321 evolving chaotically near 1 + 1U 2N with resonance’s strength Dq ’ 0:007. Mean a calculated with a running window of 50,000 yrs.

circular orbits Dq /ðsin iÞq for even order resonances and occur in a wide variety of orbital configurations of the three bodies. Dq /ðsin iÞ2q for odd order resonances. That means, our algorithm Then, in a migration scenario in a protoplanetary disk, bodies in predicts that the lower order terms in the disturbing function for quasi circular orbits could be trapped in zero order TBRs since circular orbits would be factorized by ðsin iÞq for q even and by these resonances have some associated strength (Quillen, 2011). ðsin iÞ2q for q odd. We have no knowledge of theoretical predictions about this type of dependence of the strength on the orbital incli- nation in the case of TBRs but we know that two body resonances 3. Atlas of the three body resonances in the Solar System have also an asymmetric behavior between eccentricity and inclination. For each pair of perturbing planets assumed in circular coplanar It is interesting to note that two body resonances and TBRs orbits, the strength Dq depends on the semimajor axis of the TBR q share the same property: their strengths are proportional to e defined by ðk0; k1; k2Þ and on the orbital elements ðe; i; xÞ of the which means that quasi circular resonant orbits are weakly massless particle. We computed Dq for all TBRs with p 6 20 bounded except for zero order resonances which are the strongest involving pairs of the 8 planets from 0 to 1000 au assuming a test ones, they are almost independent of the eccentricity and have particle with e ¼ 0:15; i ¼ 6, which is typical of the main asteroid some strength even for zero eccentricity orbits. But there is a fun- belt and an arbitrary x ¼ 60. The general view of the distribution damental difference: zero order two body resonances are confined of these 55,814 resonances is showed in Figs. 11 and 12 which can to co-orbital motions (trojans, for example) while zero order TBRs be qualitatively compared with the atlas of two body resonances 284 T. Gallardo / Icarus 231 (2014) 273–286

67.9 2003 QK91 67.85

67.8

67.75

67.7 mean a (au) 67.65

67.6

270

180 (27-8N)

σ 90

360

270

180

(10-1U-1N) 90 σ

0 0 0.1 0.2 0.3 0.4 0.5 time (Myrs)

Fig. 20. Evolution of 2003 QK91 dominated by the two-body resonance 27-8 N (mid panel) and by the TBR 10-1U-1N (bottom panel) with strength Dq ’ 4 105. Mean a computed using a running window of 10,000 yrs. Dashed line indicates the nominal 27-8 N and the continuous line indicates the nominal 10-1U-1N.

55.14

55.12

55.1

55.08 mean a (au) 55.06 2005 CH81

270

180

90 (10+1U-6N) σ 0 0 0.1 0.2 0.3 0.4 0.5 time (Myrs)

Fig. 21. Evolution of 2005 CH81 near resonance 10 + 1U-6N. Mean a computed using a running window of 10,000 yrs. The nominal resonance is indicated with the line and its strength is 1:4 105. presented in Fig. 7 of Gallardo (2006), but no quantitative compar- respectively. Finally, for a > 250 au it appears the series of unex- ison can be done because our Dq is not exactly the amplitude of the pectedly strong resonances 1 + 1U 2N, 2 + 1U 2N, 3 + 1U 2N disturbing function of the TBRs and the scales are different. and so ones. In Tables 1 and 2 we present some of the strongest resonances We remark that our method ignores the planetary eccentricities along the Solar System up to 1000 au where resonances involving then we expect the effective strengths in the real Solar System will Jupiter and Saturn dominate except in the TNR where the reso- be greater. Moreover, as our method ignores variations in longi- nances involving Uranus and Neptune dominate instead. Table 3 tudes of the perihelia and nodes, each resonance will be composed shows the strongest resonances involving terrestrial planets with by a multiplet that our method cannot discern. jovian planets and Table 4 shows the strongest resonances involv- ing only terrestrial planets. 3.1. Density of TBRs along the Solar System There is a very dense region of TBRs between 0.5 a 2 au mainly due to resonances involving Venus–Earth, Earth–Jupiter, Venus- From the inspection of Table 1 we conclude that the strongest Jupiter and Earth–Mars. For example, the strong resonances at TBRs in the main asteroid belt have Dq 103, then to find how a 0:8 and a 1.1 au are due to 8-7V + 1J and 7 6E 1J respec- the dynamically relevant TBRs are distributed in the Solar System tively. Between 2 and 4 au there are some known strong reso- we have calculated the number of TBRs with Dq > 105 between nances involving Jupiter and Saturn immerse in a region of low intervals of 0.1 au from 0 to 40 au showing the result in Fig. 13. density of resonances. Between 4 and 35 au there is a region with As the superposition of TBRs is associated with chaotic dynamics high density of strong resonances involving the jovian planets. (Murray and Holman, 1997; Murray et al., 1998; Nesvorny´ and From 35 to 200 au resonances are weaker with the exception of Morbidelli, 1998, 1999; Morbidelli and Nesvorny´ , 1999), a larger 2 + 1S-3U and 3 + 1S-3U at a 109:5 and a 143.4 au density implies a more chaotic dynamics. Fig. 13 can be considered T. Gallardo / Icarus 231 (2014) 273–286 285 as a global indicator of the chaos generated by the TBRs in the Solar Morbidelli (1998) and Smirnov and Shevchenko (2013) found hun- System, but to make an unequivocal diagnosis of chaos it is neces- dreds of asteroids evolving in all them except in the last one. A sary to know the widths of the resonances expressed in au, point comparison with Fig. 1 from Morbidelli and Nesvorny´ (1999) will that our method in its present form cannot resolve for now. The also help to identify the peaks in the histogram. Our method indi- highest peak is between 0.7 and 1.0 au and is produced mainly cates that the strongest TBR in the asteroid belt is 1-3J + 2S (see by TBRs including Venus as the innermost planet and the Earth also Table 1) which is the second strongest resonance according as the innermost planet. It is evident that the main asteroid belt to Nesvorny´ and Morbidelli (1999) and one of the three most pop- is located in the region with lowest density of TBRs between the ulated resonances according to Smirnov and Shevchenko (2013). planets. To appreciate with more detail the situation in the aster- oids’ region we plotted in Fig. 14 a detail of Fig. 13 jointly with an histogram of the osculating semimajor axes of the asteroids ta- 3.3. Identifying a resonance: the case of 2009 SJ18 ken from ASTORB database (ftp://ftp.lowell.edu/pub/elgb/as- torb.html). It is suggestive that the stable population of asteroids We illustrate the use of the method with the case of the NEA is precisely located in the region with lowest density of TBRs in 2009 SJ18. A numerical integration of this object shows that its the Solar System. At both sides of the main asteroid belt there is semimajor axis oscillates between 0.945 and 0.950 au. Taking the a clear increase in the density. present orbital elements e; i; x (precise values are not necessary) Considering only resonances with Dq > 105, at the left of the for 2009 SJ18 we calculate Dq for all TBRs with p 6 20 located be- asteroid belt between 1.5 and 2.0 au the most common relevant tween these values of semimajor axis. The strongest resonances are resonances involve Mars–Jupiter (47%) and Earth–Jupiter (25%), in- showed in Fig. 16 where 1 1E 1J dominates by two orders of side the asteroid belt between 2.0 and 3.3 au involve mostly Jupi- magnitude with respect to its neighbors. Then, we calculate the ter–Saturn (36%), Mars–Jupiter (27%) and Earth–Jupiter (21%) and time evolution of the critical angle r ¼ k kE kJ þ - and we plot at the right of the asteroid belt between 3.3 and 4.0 au the most aðtÞ and rðtÞ at top and middle panels in Fig. 17 respectively. In common resonances involve Jupiter–Saturn (51%) and Jupiter– spite of being very close to Earth, it survives 4000 years captured Uranus (23%). in the resonance 1 1E 1J at a ¼ 0:9475 au. The correlation be- tween aðtÞ and rðtÞ confirms the object is inside the resonance, but to discard a casual coincidence we also show at bottom 3.2. Signatures of TBRs in the main asteroid belt panel in Fig. 17 the time evolution of the critical angle

r ¼ 9k 6kV kN 2- corresponding to the closest resonance to We can associate details of the distribution of asteroids to two 1 1E 1J seen in Fig. 16. The circulation of this critical angle body and TBRs as showed in Fig. 15 where it was plotted an indicates that, in spite of this resonance being located very near histogram of the synthetic proper semimajor axes computed to the value of the asteroid’s semimajor axis, the asteroid is not numerically given by AstDyS database (http://www. evolving under its influence, as we could have deduced from its hamilton.dm.unipi.it/astdys/) using bins of 0.001 au. The peaks negligible strength showed in Fig. 16. The critical angle corre- are due to concentrations generated by two body and TBRs. Note sponding to the resonance 7 6E 3M, the second strongest res- for example the strong peak at a ’ 2:419 au due to the exterior onance in the interval analyzed, also circulates. This procedure resonance 2–1M (Gallardo, 2007). By simple inspection of Fig. 15 consisting in calculating the strengths Dqðe; i; xÞ, being ðe; i; xÞ we find peaks in the histogram where two body resonances are the orbital elements of the asteroid we are studying, for all reso- absent but TBRs are present like 1-4J + 2S at 2.397 au, 1-4J + 3S at nances with nominal location near the asteroid’s semimajor axis, 2.622 au, 1-3J + 1S at 2.752 au, 2-5J + 2S at 3.173 au and 3-7J + 2S looking for the strongest ones and calculating their critical angles, at 3.207 au. All these were already studied by Nesvorny´ and becomes very useful to identify the TBRs affecting the asteroid, if

15.94

15.9 Chariklo

15.86

15.82 mean a (au)

15.78 360

270

180 (3-4U) σ 90

360

270

180

(5-2S-2N) 90 σ

0 -0.1 -0.05 0 0.05 0.1 time (Myrs)

Fig. 22. Centaur 10199 Chariklo in the past appears captured in the resonance 3-4U at a ¼ 15:87 au (mid panel) but at t’23,800 yrs its semimajor axis drops very close to the nominal value of the TBR 5 2S 2N at a ¼ 15:771 au (bottom panel) with strength Dq ’ 4 104 being captured in this resonance since then. The horizontal line in top panel indicates the location of the nominal TBR. Mean a over 2000 yrs. 286 T. Gallardo / Icarus 231 (2014) 273–286 there is any. Other NEAs evolving in resonances involving terres- eccentricity orbits it is proportional to ðsin iÞz with z ¼ q for even trial planets can be detected by a similar procedure. order TBRs and z ¼ 2q for odd order TBRs. The algorithm allows to identify the strongest TBRs near a given semimajor axis for a given set of elements ðe; i; xÞ of the test par- 3.4. Some cases of TNOs and centaurs ticle and following this procedure we obtained an atlas of the strongest TBRs in the Solar System. The main asteroid belt is We call the attention to the resonances 1 + 1U 2N, crossed by several strong TBRs involving Jupiter and Saturn that 2+1U 2N, 3 + 1U 2N and so ones in the far TNR. They are appear as peaks in an histogram of proper semimajor axes, but unusually strong in a region where other TBRs are several orders immerse in a region with the lowest density of TBRs in the Solar of magnitude weaker. In order to appreciate the dynamical effects System. Both borders of the asteroid belt are characterized by an of these TBRs we numerically integrated a set of particles with ini- increase in the density of relatively strong TBRs reinforcing the tial semimajor axes a 6 a0 being a0 the nominal location of the res- claims that TBRs have a relevant role in the stability of that region. onance 1 + 1U 2N and imposing a continuous perturbation so We also found a series of strong TBRs of the type n + 1U 2N with that the particles’ semimajor axes slowly increase with time and n integer in the far TNR, specially at 262; 416 and 545 au. As eventually cross or get trapped into the resonance. We considered illustration we showed some objects in TBRs involving other plan- only the planets Uranus and Neptune in coplanar circular orbits in ets than Jupiter and Saturn. This paper it is not a systematic survey order to make the experiment under the hypotheses of our algo- of objects in TBRs, we are confident that if a survey of this kind is rithm. The dynamical effect of the resonance in the semimajor axis done with the help of our method it will identify several minor is clearly seen in Fig. 18 which shows the evolution of two parti- bodies in TBRs involving pairs of planets other than Jupiter–Saturn. cles, one that crosses the resonance and other that started inside The complete atlas and codes for computing qð Þ and Dq are and remains captured overcoming the forced migration with the r available under request to the corresponding author. resonance’s strength which in this experiment is Dq ’ 0:014. Regarding the actual population of TNOs, 2006 UL321 has a very Acknowledgments preliminary orbit determination, but taking their nominal ele- ments from JPL (ssd.jpl.nasa.gov/sbdb_query.cgi) we obtained an Two anonymous reviewers contributed to improve substan- orbital evolution near 1 + 1U 2N with Dq ’ 0:007 showed in tially the original manuscript. Partial support from PEDECIBA is Fig. 19. We performed numerical integrations of 87269 (2000 acknowledged. OO67) with clones and all them resulted evolving chaotically near the resonance 3 + 1U 2N. Other objects evolving in resonances References with Uranus and Neptune are for example Eris, which is evolving very near the resonance 10-1U-1N (Gallardo et al., 2012) with Aksnes, K., 1988. General formulas for three-body resonances. NATO Advanced 5 strength 3 10 , 2003 QK91 which is also evolving in that reso- Study Institute on Long-Term Dynamical Behaviour of Natural and Artificial N- nance with similar strength but with a strong influence of the Body Systems, pp. 125–139. two-body resonance 27-8N as shown in Fig. 20 and 2005 CH81 Cachucho, F., Cincota, P.M., Ferraz-Mello, S., 2010. Chirikov diffusion in the asteroidal three-body resonance (5, 2, 2). Celest. Mech. Dynam. Astron. which is evolving near the resonance 10 + 1U-6 N as shown in 108, 35–58. Fig. 21 with strength 1:4 105. de La Fuente Marcos, R., de La Fuente Marcos, C., 2008. Confined chaotic motion in A backwards integration shows the centaur 10199 Chariklo was three-body resonances: Trapping of trans-Neptunian material induced by gas- drag. Mon. Not. R. Astron. Soc. 388, 293–306. captured in the resonance 3-4U at a = 15.87 au, but at present is Fernández, J.A., Gallardo, T., Brunini, A., 2002. Are there many inactive Jupiter family outside this resonance in a region where, according to its orbital comets among the near-Earth asteroid population? Icarus 159, 358–368. elements, TBRs 5 2S 2N at 15.771 au, 9-1J-5U at 15.774 au Gallardo, T., 2006. Atlas of mean motion resonances in the Solar System. Icarus 184, 4 29–38. and 7-4S + 2U 15.776 au dominate with strengths 4 10 ; Gallardo, T., 2007. The Mars 1:2 resonant population. Icarus 190, 280–282. 4 5 7 10 and 9 10 respectively. Fig. 22 shows the critical angle Gallardo, T., Hugo, G., Pais, P., 2012. Survey of Kozai dynamics beyond Neptune. for the resonance 3-4U in mid panel and the critical angle for the Icarus 220, 392–403. Gomes, R.S., Gallardo, T., Fernández, J.A., Brunini, A., 2005. On the origin of the high TBR 5 2S 2N in the low panel. The critical angle for 9-1J-5U perihelion scattered disk: The role of the Kozai mechanism and mean motion circulates and the critical angle for 7-4S + 2U slowly circulates. resonances. Celest. Mech. Dynam. Astron. 91, 109–129. Finally, we want to stress that in our numerical integrations of Guzzo, M., 2005. The web of three-planet resonances in the outer Solar System. Icarus 174, 273–284. real and fictitious objects in the outer Solar System and in the TNR, Michtchenko, T.A., Malhotra, R., 2004. Secular dynamics of the three-body problem: when looking for TBRs we found very common also the capture in application to the t Andromedae planetary system. Icarus 168, 237–248. high order two body resonances with Neptune or Uranus as was Morbidelli, A., 2002. Modern Celestial Mechanics. Aspects of Solar System the case of Chariklo and 2003 QK91. The interaction between TBRs Dynamics. Taylor and Francis. Morbidelli, A., Nesvorny´ , D., 1999. Numerous weak resonances drive asteroids and high order two body resonances seems to be a common toward terrestrial planets orbits. Icarus 139, 295–308. dynamical situation. Murray, N., Holman, M., 1997. Diffusive chaos in the outer belt. Astron. J. 114, 1246– 1259. Murray, N., Holman, M., Potter, M., 1998. On the origin of chaos in the Solar System. 4. Conclusions Astron. J. 116, 2583–2589. Nesvorny´ , D., Morbidelli, A., 1998. Three-body mean-motion resonances and the chaotic structure of the asteroid belt. Astron. J. 116, 3029–3037. We defined a function qðrÞ related to the resonant disturbing Nesvorny´ , D., Morbidelli, A., 1999. An analytic model of three-body mean motion function which allows us to estimate the strengths of arbitrary resonances. Celest. Mech. Dynam. Astron. 71, 243–271. Okyay, T., 1935. Über die mehrfachen Kommensurabilitäten im system planetoid- TBRs with no restriction for the orbital elements of the particle’s Jupiter–Saturn. Astron. Nachrichten 255, 277–300. orbit but taking circular and coplanar orbits for the two planets. Quillen, A.C., 2011. Three-body resonance overlap in closely spaced multiple-planet Our results are roughly in agreement with previous theoretical systems. Mon. Not. R. Astron. Soc. 418, 1043–1054. Schubart, J., 1968. Long-period effects in the motion of hilda-TYPE Planets. Astron. J. and numerical studies of TBRs involving Jupiter–Saturn for the pla- 73, 99–103. nar case and predict some new results. For example, our algorithm Smirnov, E.A., Shevchenko, I.I., 2013. 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